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A portal vector-like lepton can make muon-collider production of a Higgs boson with a dark photon (hγd) outrun hZ production by a factor of 1–100, giving a kinetic-mixing-independent dark-photon probe.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-04 00:31 UTC pith:IOTNXO2H

load-bearing objection Good first paper on a genuinely new hγd channel at a muon collider; printed Eq. (25) is a typo that must be fixed, and the abstract's DM relic-density claim is unsupported — but the core physics holds up and deserves referee time. the 2 major comments →

arxiv 2511.00578 v3 pith:IOTNXO2H submitted 2025-11-01 hep-ph hep-ex

Implications of portal vector-like lepton on associated Higgs production at a multi-TeV muon collider

Krishna Tewary , Sanjoy Biswas , Shivam Verma This is my paper
classification hep-ph hep-ex
keywords portal vector-like leptonmuon colliderdark photonHiggs associated productionnon-decoupling effectmuon g-2missing energyjet substructure
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the Standard Model with a heavy vector-like lepton that carries dark-sector charge and mixes with the muon. Its central claim is that at a multi-TeV muon collider, μ+μ− → hγd is strongly enhanced by heavy-lepton exchange, while μ+μ− → hZ stays at its Standard Model rate; the ratio can reach 1–100 within perturbative unitarity. The authors find that this process probes dark photon masses of 10–100 GeV and the dark gauge coupling, independent of the usual kinetic-mixing parameter, and they present a b-bbar plus missing-energy analysis reaching 2σ exclusion up to mγd ≈ 80 GeV. If right, the channel turns a would-be background into a new way to study Higgs production and the dark sector at a future muon collider.

Core claim

Using a portal vector-like lepton (an SU(2)-singlet lepton with hypercharge −1 and a dark U(1)D charge, mixing only with the muon), the paper shows that the right-handed muon–heavy-muon–dark-photon coupling survives the mass-basis rotation, while the corresponding Z coupling cancels because the muon and its heavy partner share the same SM charges. This allows t/u-channel exchange of the TeV-scale heavy lepton to give an hγd cross section of roughly a few fb to 10^2 fb, exceeding hZ by 1–100 even for sinθL as small as 10^-6–10^-5, provided the right-handed mixing angle sinθR reaches O(0.1) as the paper argues. The hZ rate remains essentially SM-like across the allowed parameter space, so the

What carries the argument

The portal Yukawa interaction ωf Φd μ'pL μ'R together with U(1)D gauging; after spontaneous symmetry breaking this produces mixing angles θL and θR that diagonalize the muon–heavy-muon mass matrix. The right-handed mixing angle is the load-bearing quantity: it generates the off-diagonal μ–μp–γd vertex, and heavy-lepton t/u-channel exchange makes hγd non-decoupling, an enhancement that hZ lacks. The collider analysis then uses a mass-drop tagger on boosted jets, a missing-energy veto, and a b-bbar invariant-mass window to extract the h(b bbar) γd(invisible) signal.

Load-bearing premise

The entire enhancement rests on the right-handed mixing angle sinθR being O(0.1) for a TeV-scale heavy muon even when sinθL is tiny; the paper's printed Eq. (25) gives sinθR ~ 10^-9 for its own benchmarks, so if that formula rather than the prose is what the model implies, the hγd signal disappears.

What would settle it

Substitute the benchmark values mµp = 1 TeV and sinθL = 4×10^-5 into Eq. (25): the formula as printed yields sinθR ~ 4×10^-9, not ~0.1. Re-deriving the mixing angles from the mass matrix of Eq. (9) and checking which value of sinθR satisfies both the mass diagonalization and the g-2 bands is a one-line numerical test that would settle whether the claimed non-decoupling enhancement exists.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • The μ+μ− → hZ rate at a muon collider remains close to the Standard Model prediction in a wide parameter range, so a measured deviation would point to physics beyond this pVLL setup.
  • The hγd channel provides an additional Higgs-production sample, potentially giving up to about 10% better precision on Higgs properties than the hZ channel.
  • Absence of a b-bbar plus missing-energy excess at a muon collider would place new constraints on mγd and gd for dark photon masses between 10 and 100 GeV, independent of kinetic mixing.
  • At a 3 TeV muon collider with 1 ab^-1, the one-jet final state gives a 2σ exclusion up to mγd ≈ 80 GeV (76 GeV for the second benchmark set); the 10 TeV, 10 ab^-1 stage extends the reach in the mγd–mµp plane.
  • A substantial part of the parameter space favored by the current muon g-2 measurement can be probed through this channel.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The numerical consistency of Eq. (25) is the clearest editorial concern: substituting the paper's own benchmarks (e.g., sinθL = 4×10^-5, mµp = 1 TeV) into the printed formula gives sinθR ~ 4×10^-9, not the O(0.1) value the prose and the g-2 bands require; a corrected mixing relation would settle whether the central mechanism survives.
  • The hγd channel effectively replaces kinetic mixing with fermion mixing as the dark-photon portal, which suggests analogous portal-lepton setups for the electron or tau could yield flavour-selective dark-photon searches at other lepton colliders.
  • Since the dominant SM backgrounds arise from hνν and Zνν, a dedicated low-mγd optimization with relaxed missing-energy cuts could extend the sensitivity below the 10 GeV range examined here.
  • The authors parametrize the cross-section dependence on gd and mγd but do not fully exploit scalar-mixing-enhanced diagrams; including dark-Higgs contributions for larger sinθs could modify the reach in the mµp–mγd plane.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. This paper studies a Standard Model extension with a U(1)_D portal vector-like lepton (pVLL) mixing with the muon, a dark photon, and a dark Higgs. It computes μ+μ−→hZ and μ+μ−→hγ_d at √s=3 and 10 TeV muon colliders, claiming that the hγ_d rate can exceed the SM hZ rate by a factor of 1–100 thanks to a non-decoupling O(0.1) right-handed mixing angle sinθ_R, even for tiny left-handed mixing sinθ_L. The model is implemented in FeynRules/MadGraph, relevant constraints (muon g-2, LFU, electroweak precision, LHC) are applied, and a bbbar-plus-missing-energy collider analysis with jet substructure is presented, yielding 2σ exclusion projections for m_γd and g_d.

Significance. If the intended value of sinθ_R is used, the proposed hγ_d channel is a novel, testable signature of portal VLLs and a kinetic-mixing-independent probe of dark photons at a multi-TeV muon collider. The paper is self-contained, gives analytic matrix elements in Appendix B, and treats the muon g-2 measurement as an external constraint rather than a fitted target. Its main weakness is a load-bearing algebraic inconsistency in the printed definition of sinθ_R; the appendices and numerical results are consistent with the large sinθ_R value, so the central claim is plausible pending correction.

major comments (2)
  1. [Sec. 3.1, Eq. (25)] Equation (25) as printed gives sinθ_R = sinθ_L sqrt(m_μ^2/m_μp^2 cos^2θ_L + sin^2θ_L) ≈ sinθ_L m_μ/m_μp ≈ 4×10^-9 for BP1, while the text, vertex (A.2), Appendix B, Table 3/Fig. 3, and Fig. 1 all require sinθ_R ≈ (m_μp/m_μ) sinθ_L ≈ 0.38. The t/u-channel amplitude carries sinθ_R in both the γ_d vertex (A.2) and the Higgs vertex (A.8), so the printed relation suppresses the rate by many orders of magnitude, eliminating the claimed enhancement. Correct Eq. (25) and state explicitly which relation was used in the FeynRules/MadGraph implementation and in all numerical results.
  2. [Sec. 2.4, Eq. (21) and Fig. 1] The muon g-2 contribution in Eq. (21) is proportional to sin2θ_R sin2θ_L. With Eq. (25) taken literally, sin2θ_R ~ 10^-9 and the allowed bands in Fig. 1 cannot be populated for any benchmark; with the corrected large θ_R they can. The paper therefore currently contains two mutually inconsistent versions of the model. Please recompute Fig. 1 and the g-2-consistent regions using the corrected formula.
minor comments (5)
  1. [Abstract] 'We have explore' should read 'We have explored'; the phrase 'aportal' needs spacing.
  2. [Abstract and Sec. 3.1] The abstract states the hγ_d/hZ ratio is 1–100, but Fig. 5 shows ratios above 100 (up to ~3.6×10^3) in the unitarity-allowed region. Please state whether '1–100' is a representative range or a hard upper bound.
  3. [Tables 5 and 7] The signal cross-sections are quoted with BR(h→bb)=1, while the significance calculation uses 57%. This is stated only in the text near Eq. (31); an explicit note in the table captions would avoid confusion.
  4. [Sec. 3.1] 'The later is suppressed' should read 'The latter is suppressed'.
  5. [Sec. 3 / Reproducibility] The FeynRules model is described but not provided; making the UFO/parameter cards publicly available would improve reproducibility.

Circularity Check

0 steps flagged

No circularity found: model predictions follow from a specified Lagrangian with externally constrained inputs; Eq. (25) is an apparent typo, not a circular step.

full rationale

The paper derives µ+µ−→hγd rates from a concrete Lagrangian (Eqs. 1–8), a mass matrix (Eq. 9), and a bi-unitary rotation (Eq. 10). The enhancement is traced to the off-diagonal µR–µpR–γd vertex Eq. (A.2), proportional to g_d sinθR cosθR, and to the t/u-channel amplitudes in Appendix B. These are fixed by the model parameters, not fitted to the predicted signal. The muon g−2 measurement is used as an external constraint to restrict parameter space (Eq. 21 and Fig. 1); no parameter is adjusted to reproduce the hγd cross section. There are no load-bearing self-citations: the cited portal-matter literature is external model motivation, not a substitute for the calculation, and no uniqueness theorem or ansatz is imported from the authors' own prior work. The only notable issue is internal to the manuscript: Eq. (25) as printed gives sinθR ≈ sinθL·mµ/mµp ∼ 10^-9 for the benchmarks, whereas the prose, the vertex factors, the cross-section tables, and the g−2 bands require sinθR ∼ O(0.1). That is a numerical/typographical inconsistency to be corrected, not a case of a prediction reducing to its input by construction; the correct diagonalization relation is determined by Eqs. (9)–(10). Therefore the derivation is self-contained against external benchmarks and no circularity is present.

Axiom & Free-Parameter Ledger

5 free parameters · 4 axioms · 1 invented entities

The model contains five free parameters; the central signal depends mainly on the combination g_d sinθ_L m_μp/m_μ. The pVLL is adopted from prior portal-matter work and scanned, not predicted. The invisible dark-photon assumption is essential to the missing-energy signature and is not backed by an explicit DM sector in the paper.

free parameters (5)
  • m_μp (pVLL mass) = 1, 2, 3 TeV in benchmarks
    Heavy muon partner mass; scanned over benchmark points, no first-principles value.
  • sinθ_L (left muon-pVLL mixing) = 4×10^-5, 2×10^-5, 1.66×10^-6
    Chosen small to satisfy lepton flavor universality and perturbative unitarity.
  • g_d (dark gauge coupling) = 0.05, 0.36, 0.45, 0.5
    Dark sector gauge coupling; scanned.
  • m_γd (dark photon mass) = 50, 90 GeV
    Scanned; related to v_d by m_γd = g_d v_d.
  • sinθ_s (scalar mixing angle) = 0.05
    Overall rescaling of Higgs couplings; set to 0.05 consistent with LHC scalar constraints.
axioms (4)
  • domain assumption pVLL mixes only with the second-generation charged lepton
    Evades lepton-flavor violation constraints (Sec. 2).
  • domain assumption Gauge kinetic mixing ε is small and the dark photon decays invisibly
    Required for the missing-energy signal; footnote 1 assumes a DM coupling, but no explicit DM particle is defined in the body.
  • standard math Perturbative unitarity bound |ω_f| < √2 min[(m_μp - m_μ)/v_d, 4√π]
    Used in Eq. (15) to restrict parameter space.
  • domain assumption SM background consists only of bbνν subprocesses
    The analysis tabulates hZ, ZZ, hνν, Zνν and non-resonant bbνν; other potential backgrounds (e.g., WW+jets) are not included.
invented entities (1)
  • Portal vector-like lepton (pVLL) µ_p no independent evidence
    purpose: Provides the muon–heavy-muon–dark-photon vertex that drives the enhanced hγd production
    The paper scans its mass 1–3 TeV and mixing angles; no mass or coupling prediction gives a falsifiable handle outside the paper's own parameter scan.

pith-pipeline@v1.3.0-alltime-deepseek · 32528 in / 21523 out tokens · 214661 ms · 2026-08-04T00:31:38.955551+00:00 · methodology

0 comments
read the original abstract

We have explore a portal vector-like lepton (pVLL) extension of the Standard Model (SM) and study its implications for Higgs and vector-boson associated production ($hV$, with $~V = Z$-boson or dark photon) at a future muon collider facility. We show that while the $~\mu^+ \mu^- \to hZ~$ production rate remains close to its SM prediction in a wide range of parameter space, the rate for $~\mu^+ \mu^- \to h\gamma_d~$ can be substantially enhanced owing to the non-decoupling nature of the interaction involving the heavy lepton, the muon and the dark photon. We demonstrate that the $h\gamma_d$ production rate can exceed the corresponding $hZ$ rate by a factor of $1$-$100$ within the perturbative unitarity limit, making it a promising channel for probing Higgs interactions and potential new physics effects. We also examine the role of the pVLL state in the context of dark matter (DM) phenomenology and identify regions of parameter space consistent with the observed relic abundance by extending the simplified setup with a viable DM candidate. The $h\gamma_d$ production can also be used to constrain the dark photon mass ($m_{\gamma_d}$) and/or the dark gauge coupling ($g_d$) consistent with various constraints including the current muon $g-2$ measurements within the pVLL framework. We perform a detailed collider analysis of the $h\gamma_d$ process in the $b\bar{b}~+$ missing energy final state. A $2\sigma$ exclusion limit for $m_{\gamma_d}$ up to $80$ GeV is obtained assuming $~g_d=0.05$, $~\sin\theta_L=4\times10^{-5}$, $~\sin\theta_s=0.05$, for a heavy lepton mass $~\sim 3$ TeV at a $3$ TeV muon collider with an integrated luminosity of $1$ ab$^{-1}$.

Figures

Figures reproduced from arXiv: 2511.00578 by Krishna Tewary, Sanjoy Biswas, Shivam Verma.

Figure 1
Figure 1. Figure 1: Region (coloured) of parameter space allowed by muon [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Feynman diagrams for associated production of Higgs and a vector boson ( [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Ratios of cross sections ( Ri ’s defined in Eq. (23) ) for different benchmark points at (a) √ s = 3 TeV and (b) 10 TeV. One can see that while the cross section for µ +µ − → hZ production in this scenario remains the same as that in the SM, a higher Higgs production rate is predicted in the µ +µ − → hγd channel. This is true even for a sin θL as small4 as ∼ 10−5 with reasonable choices of other relevant p… view at source ↗
Figure 4
Figure 4. Figure 4: Variation of ratio of cross sections (R2) for the process µ +µ − → hγd with respect to the SM hZ production rate in mγd –mµp plane assuming √ s = 3 TeV for two different set of (gd, sin θL) and a fixed value of sin θs. (a) The left plot corresponds to gd = 0.05, sin θs = 0.05, and sin θL = 4 × 10−5 and (b) the right plot corresponds to gd = 0.5, sin θs = 0.05, and sin θL = 1.66 × 10−6 . The region above th… view at source ↗
Figure 5
Figure 5. Figure 5: Variation of ratio of cross sections (R2) for the process µ +µ − → hγd with respect to the SM hZ production rate in mγd –mµp plane assuming √ s = 10 TeV for two different set of (gd, sin θL) and a fixed value of sin θs. (a) The left plot corresponds to gd = 0.05, sin θs = 0.05, and sin θL = 4 × 10−5 and (b) the right plot corresponds to gd = 0.5, sin θs = 0.05, and sin θL = 1.66 × 10−6 . The region above t… view at source ↗
Figure 6
Figure 6. Figure 6: LO cross section for µ +µ − → hγd as a function of the muon collider center-of-mass energy for various choices of the mµp and sin θL assuming (a) sin θs = 0.05, gd = 0.36, mγd = 50 GeV and (b) sin θs = 0.05, gd = 0.50, mγd = 90 GeV. The LO cross sections for the process µ +µ − → hγd for various benchmark points mentioned in [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Feynman diagrams for the subprocess, µ +µ − → h(b ¯b) + γd leading to b ¯b plus missing energy final state. The dominant SM backgrounds that contribute to the b ¯b plus missing energy final state, namely, µ +µ − → b ¯bνν¯ process comes from hZ, ZZ, hνν, Zν ¯ ν¯ and non-resonant b ¯bνν¯ productions at muon collider. They can be sub-divided into 2 → 2, 2 → 3 and 2 → 4 categories as follows µ +µ − → hZ → (b ¯… view at source ↗
Figure 8
Figure 8. Figure 8: Feynman diagrams for various SM background subprocesses leading to [PITH_FULL_IMAGE:figures/full_fig_p013_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Central-jet (Nc−jet) multiplicity distribution at √ s = 3 TeV and 10 TeV for three different signal benchmark points and the total SM background. −1 0 1 2 3 4 Nc−jet 0.0 0.2 0.4 0.6 0.8 1.0 Normalized Events √s = 3 TeV b¯bνν¯ (total) b¯bνν¯(Zh) b¯bνν¯(ZZ) b¯bνν¯(hνν¯) (a) −1 0 1 2 3 4 Nc−jet 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Normalized Events √s = 10 TeV b¯bνν¯ (total) b¯bνν¯(Zh) b¯bνν¯(ZZ) b¯bνν¯(hνν¯) (b) [PI… view at source ↗
Figure 10
Figure 10. Figure 10: Central-jet (Nc−jet) multiplicity distribution at √ s = 3 TeV and 10 TeV for various SM background subprocesses and the total background. This behavior arises because a significant fraction of the background originates from 2 → 3 processes such as hνν¯ and Zνν¯ 7 . In these cases, the Higgs or Z boson produced in association with a neutrino pair is generally not boosted, often resulting in multi-jet final… view at source ↗
Figure 11
Figure 11. Figure 11: Transverse momentum (pT ) distributions of the (a) jet and (b) the di-jet in the Nc−jet = 1 and Nc−jet = 2 final states, respectively, at 3 TeV muon collider center-of-mass energy for both signal and the SM background events. 0 1000 2000 3000 4000 5000 6000 pT (jet) [GeV] 0.0 1.0 2.0 3.0 4.0 5.0 Normalized Events ×10−3 Nc−jet=1 √s = 10 TeV BP1 BP2 BP3 b¯bνν¯ (total) b¯bνν¯(Zh) b¯bνν¯(ZZ) b¯bνν¯(hνν¯) (a) … view at source ↗
Figure 12
Figure 12. Figure 12: Transverse momentum (pT ) distributions of the (a) jet and (b) the di-jet in the Nc−jet = 1 and Nc−jet = 2 final states, respectively, at 10 TeV muon collider center-of-mass energy for both signal and the SM background events. As already mentioned, Higgs boson produced at the future muon collider center-of-mass energies in a 2 → 2 process are typically highly boosted. The Higgs subsequently decays into a … view at source ↗
Figure 13
Figure 13. Figure 13: Missing energy (E/ ) distributions at √ s = 3 TeV in the (a) Nc−jet = 1 and (b) Nc−jet = 2 final states for both signal and SM background events. 4000 5000 6000 7000 8000 9000 10000 11000 Missing Energy (E/ ) [GeV] 0.0 0.2 0.5 0.8 1.0 1.2 1.5 1.8 2.0 2.2 Normalized Events ×10−3 Nc−jet=1 √s = 10 TeV BP1 BP2 BP3 b¯bνν¯ (total) b¯bνν¯(Zh) b¯bνν¯(ZZ) b¯bνν¯(hνν¯) (a) 4000 5000 6000 7000 8000 9000 10000 11000 … view at source ↗
Figure 14
Figure 14. Figure 14: Missing energy (E/ ) distributions at √ s = 10 TeV in the (a) Nc−jet = 1 and (b) Nc−jet = 2 final states for both signal and SM background events. • b¯b invariant mass (Mb¯b ) : In our analysis the invariant mass associated with the b ¯b system is 18 [PITH_FULL_IMAGE:figures/full_fig_p018_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Invariant mass (Mb¯b ) distribution at √ s = 3 TeV in the (a) Nc−jet = 1 and (b) Nc−jet = 2 final state for both signal and background events. 19 [PITH_FULL_IMAGE:figures/full_fig_p019_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Invariant mass (Mb¯b ) distribution at √ s = 10 TeV in the (a) Nc−jet = 1 and (b) Nc−jet = 2 final state for both signal and background events. One can see that the normalized signal distributions are mostly insensitive to the intermediate heavy muon mass. While presenting the kinematic distributions for the signal benchmark points (BPs) we have chosen (BP1, BP2 and BP3) as our representative BPs correspo… view at source ↗
Figure 17
Figure 17. Figure 17: 2σ exclusion limit in the mγd−mµp plane for two different final states at 3 TeV muon collider center-of-mass energy for (a) gd = 0.05, sin θL = 4 × 10−5 , sin θs = 0.05 and (b) gd = 0.5, sin θL = 1.66 × 10−6 , sin θs = 0.05 assuming an integrated luminosity of 1 ab−1 . The exclusion limits in Nc−jet = 1 (pink) and Nc−jet = 2 (blue) final states are represented by the region below the dash-dot lines. The r… view at source ↗
Figure 18
Figure 18. Figure 18: 2σ exclusion limit in the mγd − mµp plane in Nc−jet = 1 final state at 10 TeV muon collider center-of-mass energy for (a) gd = 0.05, sin θL = 4 × 10−5 , sin θs = 0.05 and (b) gd = 0.5, sin θL = 1.66 × 10−6 , sin θs = 0.05 assuming an integrated luminosity of 10 ab−1 . The 2σ exclusion limit is represented by dash-dot line (pink). The region above the dashed (black) line is allowed by the constraint repres… view at source ↗
Figure 19
Figure 19. Figure 19: 2σ exclusion limit (represented by dashed-dot line) in the gd − mγd plane using the data corresponding to √ s = 3 TeV and an integrated luminosity of 1 ab−1 in the Nc−jet = 1 (pink) and Nc−jet = 2 (blue) final states for two different sets of sin θL and mµp : (a) sin θL = 4 × 10−5 , mµp = 1 TeV and (b) sin θL = 1.66 × 10−6 , mµp = 3 TeV. The region above the dashed (black) line is allowed by the constrain… view at source ↗
Figure 20
Figure 20. Figure 20: Feynman diagrams for the process µ +µ − → h γd mediated by the portal matter mµp through the (a) t-channel and (b) u-channel. The amplitude for the t-channel diagram is given by iMt = −i " v¯s2 (k2) γ µ [PITH_FULL_IMAGE:figures/full_fig_p027_20.png] view at source ↗

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Forward citations

Cited by 2 Pith papers

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